Question

Intercept

- A friend of yours wants to teach a dance class after school. She needs to rent a
dance studio to teach the class, and the cost of rent is $400 for 12 weeks. The class
will meet each week and your friend will charge each student the same fee for the
whole course. Your friend's profit depends on how many students enroll in the
course and how much she charges each student.
(a) If your friend charges each student D dollars for the course, define a
function, p, that relates the number of students who take the dance course, n,
to the profit, p(n).
(b) Suppose your friend teaches 11 students and makes a profit of $425. How
would you represent this information using function notation?
(c) Use your answer to part b to solve for the amount each student is charged for
the dance course.

Answer 1

a) p(n) = D * n - 400

b) p(11) = 425

c) D = $75

Explanation:

a) To create the function, let's start with the amount of money she makes from the students. Since there are n students and each student pays D dollar, this would be the product of the two, D * n.

In addition, rent costs $400, so to calculate the total profit, 400 will have to be subtracted from what the students pay. Putting this in function notation yields:

p(n) = D * n - 400

b) In the function created in the previous step, n represents number of students and p(n) represents the function defined evaluated on the n number of students. Similarly, p(11) represents when the function in evaluated on 11 students.

Remember that the function is used to calculate the profit, which we are given to be equal to $425, meaning we can set that equal to the result of the function when n = 11:

p(11) = 425

c) Using the definition created in step a, we can show the evaluation of the product function:

p(11) = D * 11 - 400

Since n, the number of students, is 11, we could plug it in as shown above. From the previous step, we also have p(11) = 425, so we can determine that:

D * 11 - 400 = p(11) = 425

D * 11 - 400 = 425

Now, all that's left is to solve for D:

D * 11 = 825

D = 75 --> $75